Abstract
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities stays constant. In this type of models, the velocity field in the Navier–Stokes equations is not solenoidal and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.
Highlights
We study the well-posedness analysis in classes of strong solutions of class-one models1 of mass transport in isothermal, incompressible multicomponent fluids
Fluid mixtures occurring in applications are often incompressible, and the limit passage reduces the stiffness of the models by eliminating the parameter which is practically infinite
The lowMach number limit leads to a type of incompressibility condition which has not yet been studied in the context of mathematical analysis for fluid dynamical equations
Summary
We study the well-posedness analysis in classes of strong solutions of class-one models of mass transport in isothermal, incompressible multicomponent fluids. The paper [3] shows that the relation (6) occurs in the limit case when the bulk free energy density of the system adopts the singular form ψ = h∞(θ, ρ) := k(θ, ρ) if ρi. The ’elliptic equation’ (9) defines a differential operator acting on a certain relative chemical potential (variable ζ , details below) This elliptic operator is linear for the pure diffusion case, but turns to nonlinear in the presence of reactions of the general form r = r (μ). Our main purpose in this article is to obtain a first result for incompressible flow problems in the multicomponent case To this aim the condition (12), which is well-known among mathematicians, provides a good starting example.
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